Vitali covering theorem

Call a family $\mathcal{V}$ of nondegenerate closed intervals a Vitali cover (a fine cover) of $E \subseteq \mathbb{R}$ when for every $x \in E$ and every $\varepsilon > 0$ there is $I \in \mathcal{V}$ with $x \in I$ and $0 < |I| < \varepsilon$.

Vitali covering theorem. If $\lambda^{*}(E) < \infty$ and $\mathcal{V}$ is a Vitali cover of $E$, then for every $\varepsilon > 0$ there are finitely many pairwise disjoint $I_1, \dots, I_N \in \mathcal{V}$ with

$\lambda^{*}\Big(E \setminus \bigcup_{k=1}^{N} I_k\Big) < \varepsilon,$

and there is a countable pairwise disjoint family $\{I_k\}_{k \ge 1} \subseteq \mathcal{V}$ with $\lambda^{*}\big(E \setminus \bigcup_{k \ge 1} I_k\big) = 0$. The finiteness of $\lambda^{*}(E)$ may be dropped for the countable form, since $\mathbb{R}$ is a countable union of bounded pieces. The same statement holds in $\mathbb{R}^n$ for closed balls, and the underlying combinatorial device is the $5r$ covering lemma: from any family of balls with uniformly bounded radii one may extract a disjoint subfamily $\mathcal{D}$ such that the balls of $\mathcal{D}$ dilated by the factor $5$ cover the union of the original family.

Not proved in this library. It is recorded with citations and used in no proof here.

What would prove it. Lebesgue outer measure and the measurability theory of Lebesgue measure and the Lebesgue integral ‡, together with a greedy selection: repeatedly choose an interval of nearly maximal length among those disjoint from the ones already chosen, and estimate the leftover by the $5r$ lemma. The selection is a recursion over a countable index set with an explicit rule, so it costs only dependent choice, not the full Axiom of Choice.

Which page it serves. It is the covering machinery behind almost everywhere differentiability: the Lebesgue differentiation theorems (Lebesgue's differentiation theorem for monotone functions ‡, Lebesgue differentiation theorem for $L^1$ functions ‡) and, through them, the sharp fundamental theorem of calculus (The sharp fundamental theorem of calculus (absolute continuity) ‡). Its natural home is the monotone functions and discontinuities page, which can prove that a monotone function has at most countably many discontinuities but cannot reach differentiability almost everywhere.

Relation to the elementary covering arguments already in scope. The Heine-Borel and nested interval arguments this library uses, and the elementary covering definition of a null set, are all finite or countable covering statements about total length. Vitali's theorem is the first covering result that needs the measure itself rather than just total length, which is why it is here and they are not. The weaker form sufficient for null sets is recorded separately as Mini-Vitali covering theorem ‡.